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Thursday, October 22, 2015

Sticky Points

A sticky point is a dot that stays in the right location on a Desmos (or GeoGebra) graph no matter how the sliders are moved. Why don't I just show you?

It came out of my new and improved way of introducing my students to making their own Desmos/GeoGebras to study functions. I decided to frustrate the heck out of them so that they would beg me to show them how to do the thing I wanted them to learn in the first place. Inspired by Dan Meyer's headache-for-which-math-is-the-aspirin idea. Getting a point to stick shall be their aspirin.

...then type in the coordinates of the function's y-intercept in order to get an orange dot to show up on the graph like so:

...then move the sliders around again, and repeat:

and repeat, and repeat....and of course, most of the time, the y-intercept changes, so they had to keep re-typing the orange dot's coordinates so that it was where the new y-intercept was. Frustration! Eventually (read - as soon as someone started whining) I said

"Wouldn't it be nice if that orange dot automatically moved to the location of the y-intercept as soon as you moved the sliders?" So the next activity slide was all about what ordered pair can we type into Desmos so that that happens:

Well, because of the way the DAB is made, they could easily share their ideas with me, and each other, about how to do that:

And voila, if the orange dot moves around and is always at the y-intercept right along with the sliders, that point is sticking. It's a sticky point.

The next slide asked that they do the same thing for the x-intercept - to type in an ordered pair that will stick to the x-intercept, no matter how we move the sliders. This took more time, of course, which on one level is amazing to me because they spent an entire year already on the linear function - how can they not be experts at finding its zero?

BUT, on the other hand, solving 4x - 2 = 0 is very different from solving ax + k = 0. The second one requires that they see the a and the k as numbers, even though they're letters, and seeing the x as a variable, even though it, too, is a letter. I'm sure the sliders sitting right there in front of them, with numerical values showing, helps with this idea. Interestingly, one student, who happens to do a lot of coding, got it immediately.

Next day followup & new activity:

The next day, I shared the graphs of those students who had everything sticking, so that everyone could have the experience of typing in the formulas and seeing that the points stick.

Which was the same idea, getting points to stick, but this time, the vertex, the y-intercept, and the zeros. The vertex was super easy and most got it right away, so now I wanted to bring up how to use the sliders to check if your point is sticking. Here were their responses to that:

After all, I want them to not only get their formulas right, I want them to be able to decide, and be their own teacher, about when they're right AND, more importantly, know when they're not.

When things got really interesting, for me anyway!

Once the vertex was sticking, it was on to the y-intercept. Here's where things got really interesting. Again, it was no problem for anyone to calculate the y-int when a, h, and k were numbers, but slow going when they were just a, h, and k. Eventually, here were their responses on the slide that prompted them to share:

They were checking with the sliders! And a few unexpected things popped up - one student mentioned the y-intercept for the general form of the quadratic, and one simplified the expression a(0-h)² + k to ah² + k. I had the opportunity to talk to those students about their particular work - the one who was thinking about the general form eventually made a whole desmos just about that, and got those same points to stick!

With a tool like this, anything is possible. It puts control, if that's the right word, in everyone's hands.

And here's the thing - You don't get a tool like the DAB, where who learns what is all up for grabs, and use it to make something with the same single outcome for every student. It's just not possible!

My ultimate goal....student-created GeoGebra's!

All of this was ultimately leading up to their first GeoGebras about the Absolute value function, which is their first new function for this year. They just started them yesterday, and there's plenty to add, but I already find it's going MUCH better than in previous years. They are already familiar with the sliders, AND with the idea of formulas for important points. I'll share those here soon, but in the meantime, some kids are already sharing them via the #ggbchat hashtag:

Brilliant. Honestly, mostly of my personal fluency with seeing structure in equations and rearranging expressions has come in the context of programming / desmosing / geogebraing, so I really like this context for the students.

What if you challenged the students to make the sticky points for roots of a(x-h)^2+k? Then switch the form to ax^2+bx+c and watch them derive the quadratic formula!

Thanks Andrew! Actually, finding the zeros in vertex form was the last part of the activity. My coder got them immediately. Most others got only one - the positive root. It was pure derivation heaven! I offered as a bonus for them to create a desmos with 4 sticky points using the general form, and a few took me up on it. These people remembered their quadratic formulas from last year though, so no need to derive it. Finally, I offered more bonus points for anyone who did one using the zeros form. Same takers, same fun!